Subgroups — #28505102

Order 15104.

370 conjugacy classes of subgroups.

The subgroup lattice has 166 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₅₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₅₉⋊C₁₆	C₂	1	4	yes	maximal
3	#21423782	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅₉×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₅₉⋊C₁₆	C₂²	1	2	yes	—
6	C₂×#14273638	C₄	1	1	yes	characteristic
7	C₂×#14273637	C₄	1	2	yes	—
8	C₂×C₈×Dic₅₉	C₂²	1	2	yes	—
9	#16837248	C₄	1	1	yes	characteristic
10	C₄⋊(C₅₉⋊C₁₆)	—	2	2	no	—
11	Dic₅₉⋊C₁₆	—	2	8	no	—
12	C₂²×C₁₆×C₅₉	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
14	C₅₉×C₄⋊C₁₆	—	2	2	no	—
15	C₂²×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
16	C₂×C₄×Dic₅₉	C₂×C₄	1	2	yes	—
17	C₂×C₅₉⋊C₁₆	—	2	4	no	—
18	C₂×C₅₉⋊C₁₆	—	2	2	no	—
19	#14273638	—	2	2	no	—
20	#14273637	—	2	4	no	—
21	C₈×Dic₅₉	D₄	1	4	yes	—
22	C₈×Dic₅₉	Q₈	1	4	yes	—
23	C₅₉⋊#95	C₈	1	2	yes	—
24	C₅₉⋊#95	C₈	1	2	yes	—
25	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
27	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₆×C₅₉	—	2	2	no	—
29	C₂×C₁₆×C₅₉	—	2	4	no	—
30	C₄×C₈×C₅₉	D₄	1	2	yes	—
31	C₄×C₈×C₅₉	Q₈	1	2	yes	—
32	C₂²×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
33	C₂²×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
34	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
35	C₂²×Dic₅₉	C₂×C₈	1	4	yes	—
36	C₂×C₈×C₅₉	—	2	2	no	—
37	C₂×C₈×C₅₉	C₄⋊C₄	1	2	yes	—
38	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
39	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₈×C₅₉	C₂×Q₈	1	2	yes	—
42	C₄²×C₅₉	C₄⋊C₄	1	2	yes	—
43	C₄²×C₅₉	M₁₆	1	2	yes	—
44	C₄×Dic₅₉	C₄⋊C₄	1	4	yes	—
45	C₄×Dic₅₉	M₁₆	1	4	yes	—
46	C₂×C₅₉⋊C₈	—	2	2	no	—
47	C₂×C₅₉⋊C₈	—	2	2	no	—
48	C₂×C₅₉⋊C₈	C₄⋊C₄	1	4	yes	—
49	C₁₆×C₅₉	—	4	2	no	—
50	C₅₉⋊C₁₆	—	4	2	no	—
51	C₂³×C₅₉	C₂²×C₈	1	1	yes	characteristic
52	C₂×C₄×C₅₉	—	2	2	no	—
53	C₂×C₄×C₅₉	—	2	2	no	—
54	C₂×C₄×C₅₉	C₄⋊C₈	1	2	yes	—
55	C₂×C₄×C₅₉	C₄⋊C₈	1	2	yes	—
56	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
57	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₅₉	C₂×M₁₆	1	2	yes	—
59	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
60	C₂×Dic₅₉	—	2	4	no	—
61	C₂×Dic₅₉	—	2	4	no	—
62	C₂×Dic₅₉	C₄⋊C₈	1	8	yes	—
63	C₈×C₅₉	—	2	2	no	—
64	C₈×C₅₉	#127	1	1	yes	characteristic
65	C₈×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
66	C₈×C₅₉	C₄⋊Q₈	1	2	yes	—
67	C₅₉⋊C₈	—	2	4	no	—
68	#12899	—	59	1	no	maximal, Sylow
69	C₂²×C₅₉	C₂×C₄⋊C₈	1	2	yes	—
70	C₂²×C₅₉	C₂×C₄⋊C₈	1	1	yes	characteristic
71	C₂²×C₅₉	#504	1	2	yes	—
72	C₂²×C₅₉	#504	1	2	yes	—
73	C₄×C₅₉	—	2	2	no	—
74	C₄×C₅₉	—	2	2	no	—
75	C₄×C₅₉	#384	1	1	yes	characteristic
76	C₄×C₅₉	#423	1	2	yes	—
77	C₄×C₅₉	#423	1	1	yes	characteristic
78	Dic₅₉	—	2	8	no	—
79	C₂×C₄⋊C₁₆	—	59	4	no	—
80	C₂×C₄⋊C₁₆	—	59	1	no	—
81	C₂×C₄⋊C₁₆	D₅₉	1	1	yes	characteristic
82	#1726	—	59	1	no	—
83	C₂×C₅₉	#1842	1	1	yes	characteristic
84	C₂×C₅₉	#2151	1	4	yes	—
85	C₂×C₅₉	#2151	1	1	yes	characteristic
86	C₂×C₅₉	#2151	1	1	yes	characteristic
87	C₂²×C₁₆	—	59	2	no	—
88	C₂²×C₁₆	C₂×D₅₉	1	2	yes	—
89	C₂×#97	—	59	2	no	—
90	C₂×#97	—	59	1	no	—
91	C₂×C₄×C₈	—	59	2	no	—
92	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
93	#376	—	59	1	no	—
94	C₄⋊C₁₆	—	2	2	no	—
95	C₄⋊C₁₆	—	118	8	no	—
96	C₄⋊C₁₆	—	118	2	no	—
97	C₅₉	#12899	1	1	yes	characteristic, complemented, Sylow
98	C₂²×C₈	—	59	2	no	—
99	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
100	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
101	C₂×C₄²	—	59	2	no	—
102	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
103	C₂×C₁₆	—	2	4	no	—
104	C₂×C₁₆	—	2	2	no	—
105	C₂×C₁₆	—	118	4	no	—
106	C₂×C₁₆	—	118	2	no	—
107	#97	—	118	4	no	—
108	#97	—	118	2	no	—
109	C₄×C₈	—	59	4	no	—
110	C₄×C₈	—	59	4	no	—
111	C₄×C₈	Dic₁₁₈	1	2	yes	—
112	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
113	#95	—	59	2	no	—
114	#95	—	59	2	no	—
115	C₂²×C₄	—	59	4	no	—
116	C₂²×C₄	C₈×D₅₉	1	1	yes	characteristic
117	C₂²×C₄	C₈×D₅₉	1	1	yes	characteristic
118	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	59	4	no	—
121	C₂×C₈	—	118	2	no	—
122	C₂×C₈	—	118	2	no	—
123	C₂×C₈	Dic₅₉⋊C₄	1	2	yes	—
124	C₂×C₈	C₂×Dic₁₁₈	1	1	yes	characteristic
125	C₂×C₈	D₄×D₅₉	1	2	yes	—
126	C₂×C₈	Q₈×D₅₉	1	2	yes	—
127	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
128	C₄²	—	59	4	no	—
129	C₄²	—	59	4	no	—
130	C₄²	#11745659	1	2	yes	—
131	C₄²	Dic₅₉⋊C₄	1	2	yes	—
132	C₁₆	—	4	2	no	—
133	C₁₆	—	236	2	no	—
134	C₂³	C₂×C₈×D₅₉	1	1	yes	characteristic
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	59	8	no	—
138	C₂×C₄	—	118	4	no	—
139	C₂×C₄	—	118	4	no	—
140	C₂×C₄	Dic₅₉⋊C₈	1	2	yes	—
141	C₂×C₄	Dic₅₉⋊C₈	1	2	yes	—
142	C₂×C₄	C₄⋊C₄×D₅₉	1	2	yes	—
143	C₂×C₄	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
144	C₂×C₄	M₁₆×D₅₉	1	2	yes	—
145	C₂×C₄	C₂×#11745659	1	1	yes	characteristic
146	C₈	—	2	2	no	—
147	C₈	—	118	4	no	—
148	C₈	#14273723	1	1	yes	characteristic
149	C₈	#14273725	1	1	yes	characteristic
150	C₈	#14273726	1	2	yes	—
151	C₂²	C₄⋊C₈×D₅₉	1	2	yes	—
152	C₂²	C₂×Dic₅₉⋊C₈	1	1	yes	characteristic
153	C₂²	#16837818	1	2	yes	—
154	C₂²	#16837820	1	2	yes	—
155	C₄	—	2	2	no	—
156	C₄	—	2	2	no	—
157	C₄	—	118	8	no	—
158	C₄	#16837271	1	1	yes	characteristic
159	C₄	#16837424	1	2	yes	—
160	C₄	#16837425	1	1	yes	characteristic
161	C₂	#21424247	1	1	yes	characteristic
162	C₂	#21425853	1	4	yes	—
163	C₂	#21425854	1	1	yes	characteristic
164	C₂	#21425856	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅₉
commutator, socle, prime radical	C₂³×C₅₉	C₂²×C₈
Fitting	C₂×C₅₉×C₄⋊C₁₆	C₂
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₅₉
hypocenter	C₅₉	#12899
LCS	G ▷ C₂³×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
97. C₅₉	#12899	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₄⋊(C₅₉⋊C₁₆)	2
C₂×Dic₅₉⋊C₁₆	2
#21423782	2
C₂×C₅₉×C₄⋊C₁₆	2
#12899	59